#111465

order	512
nilpotency class	3
isoclinism	isoclinic class of #97058
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1176 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

(9,12)(13,14,17,22,18,21,25,28)(15,20,23,27,16,19,24,26);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,16)(14,19,21,20)(17,24,25,23)(22,27,28,26);(2,4)(5,7)(6,9)(8,11)(10,12)(13,16,18,15)(14,20,21,19)(17,23,25,24)(22,26,28,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=ei,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18661);